Fast Help!

I have to complete my homework in 2 days and this problem I found most difficult:

Let ABC be a right triangle with hypotenuse BC. Suppose that M is the midpoint of BC and H is the feet of the perpendicular dropped from A onto BC. A point P, distinct from A, is chosen on the opposite ray of ray AM. Let the line through H perpendicular to AB intersect PB at Q; and let the line through H perpendicular to AC meet PC at R. Prove that A is the orthocenter of triangle PQR.

I have to complete my homework in 2 days and this problem I found most difficult:

Let ABC be a right triangle with hypotenuse BC. Suppose that M is the midpoint of BC and H is the feet of the perpendicular dropped from A onto BC. A point P, distinct from A, is chosen on the opposite ray of ray AM. Let the line through H perpendicular to AB intersect PB at Q; and let the line through H perpendicular to AC meet PC at R. Prove that A is the orthocenter of triangle PQR.

See diagram. By definition of the Orthocentre to prove that is the
orthocentre of triangle it is sufficient to prove that extended
is a normal to , and that extended is a normal to .

Now it is sufficient to prove that in general extended is a normal ,
as by an equivalent argument will be a normal to .

Now the best way to proceed as far as I can see is to introduce
coordinates, and use coordinate geometry to show that angle
in the diagram is a right angle.

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<<You will need to check the algebra in this carefully>>

Let be the origin , be and be .

Then is .

The slope of is , so the slope of is , in fact
is the line . The line has equation ,
so is the point of intersection of:

,

and

.

Which is the point .

Now lies on , and may be written as for some.

To finish find the equation on the line through and , and from that
find the coordinates of . Find the slope of the line through and ,
which should be minus the slope of the line through and .

Sorry,but do you have another solution that just using similar/congruent triangles, circles...etc...I haven't learn about your way to solve this, so I don't understand that much, and I think my teacher has another more simple solution